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# Journal of Combinatorics

## Volume 9 (2018)

### Number 4

### On sequences covering all rainbow $k$-progressions

Pages: 739 – 745

DOI: https://dx.doi.org/10.4310/JOC.2018.v9.n4.a9

#### Authors

#### Abstract

Let $ac(n, k)$ denote the smallest positive integer with the property that there exists an $n$-colouring $f$ of $\{ 1, \dotsc , ac(n, k) \}$ such that for every $k$-subset $R \subseteq \{ 1, \dotsc , n \}$ there exists an (arithmetic) $k$-progression $A$ in $\{ 1, \dotsc, ac(n, k) \}$ with $\{ f(a) : a \subset A \} = R$.

Determining the behaviour of the function $ac(n, k)$ is a previously unstudied problem. We use the first moment method to give an asymptotic upper bound for $ac(n, k)$ for the case $k = o(n^{1/5})$.

#### Keywords

rainbow arithmetic progression, colouring, covering, arithmetic progression, probabilistic method, universal sequence

Leonardo Alese and Stefan Lendl acknowledge the support of the Austrian Science Fund (FWF): W1230, Doctoral Program “Discrete Mathematics”.

Stefan Lendl acknowledges the support of SFB-Transregio 109 “Discretization in Geometry & Dynamics” funded by DFG and FWF (I 2978).

Received 15 February 2018

Published 7 December 2018